Articles of measure theory

An inequality about sequences in a $\sigma$-algebra

Let $(X,\mathbb X,\mu)$ be a measure space and let $(E_n)$ be a sequence in $\mathbb X$. Show that $$\mu(\lim\inf E_n)\leq\lim\inf\mu(E_n).$$ I am quite sure I need to use the following lemma. Lemma. Let $\mu$ be a measure defined on a $\sigma$-algebra $\mathbb X$. If $(E_n)$ is an increasing sequence in $\mathbb X$, then $$\mu\left(\bigcup_{n=1}^\infty E_n\right)=\lim\mu(E_n).$$ […]

How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam’s original argument about measure theory and measurable cardinals. Here is the result I am looking for: The smallest cardinal $\kappa$ that admits a non-trivial countably-additive two-valued measure must be inaccessible. The original paper can be found below […]

Uniform Integrability

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that $$ \displaystyle \sup_{i} \int_X f_i(x) \mu(dx) < \infty$$ Under which additional conditions I can say that they are Uniformly Integrable? The sequence I have is general, of […]

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let $\Omega=\{1,2,3\}, \mathcal{A}=\mathcal{P}(\Omega)$ and $P(\{\omega\})=\tfrac{1}{3}$ for each $\omega \in \Omega$. Define a stochastic process $(X(t):t\ge 0)$ by $X(t)(\omega) = \max\{t-\omega,0\}.$ Then the filtration generated by the stochastic […]

How to understand joint distribution of a random vector?

Given a random vector, what are the domain, range and sigma algebras on them for each of its components to be a random variable i.e. measurable mapping? Specifically: is the domain of each component random variable same as the domain of the random vector, and are the sigma algebras on their domains also the same? […]

Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals .

I’m trying to prove that cardinality of Borel sets is $c$ without using the concept of ordinal number. I know that the Cardinal of Borel sets are greater than $c$ because of every point in $\mathbb R$ is Borel set, therefore it’s enough to show that the cardinal of borel sets is at most $c$ […]

Limit of $n-1$ measure of the boundary of a sphere

The measure of a sphere of radius $R$ centered in $0_{\mathbb{R}^n}$ in $\mathbb{R}^n$ is \begin{array}{l l}\int_{B_0(R)}dx_1\ldots dx_n & =\int_0^R\rho^{n-1}d\rho \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{n-1}{\varphi_1}d\varphi_1\ldots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos{\varphi_{n-1}}d\varphi_{n-1}\int_{0}^{2\pi} d\theta \\ & =\omega_n \int_0^R\rho^{n-1} d\rho \end{array} where $\omega_n$ is the $n-1$ measure of the boundary of the sphere. We can calculate the integral of the function $e^{-||x||^2}$ over $\mathbb R^n$ as $$\int_{\mathbb{R}^n}e^{-||x||^2}dx_1\ldots dx_n=\int_{\mathbb R^n}e^{-x_1^2-\ldots-x_1^n}dx_1\ldots […]

Unbounded measurable set with different inner and outer measures

I’m working on providing a counterexample to the claim that A unbounded set $A \subset \mathbb{R}$ is Lebesgue measurable if and only if its inner and outer measures are equal. Further, if $B$ is an unbounded measurable set that contains $A$, then $A$ is measurable if and only if it divides $B$ cleanly. Let me […]

Calculate an integral in a measurable space

Let $(X,\mathcal{M})$ a measurable set with measure $\mu$. Let $f$ be an integrable non negative function, such that $K:=\int_{E}f \mathrm d\mu<\infty$, where $E\in(X,\mathcal M)$. Let $\alpha>0$. Calculate the following integral $\displaystyle \lim_{n\rightarrow\infty}\int_{E} n \ln\left(1+\left(\frac{f}{n}\right)^{\alpha}\right)\mathrm{d}\mu$ I prove that for $\alpha=1$ the previous integral is $K$, and $\forall \alpha>1$ is zero. It follows by the identity $(1+x^{\alpha})\leq\alpha […]

Is the countable union of measure-zero sets zero?

Let $E=\cup_{i=1}^\infty A_i$ where the measure of $A_i$ is zero. How can I conclude that $E$ has measure zero?